Probing Dynamics of Oligosaccharides by Interference Phenomena in NMR Relaxation

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Probing Dynamics of Oligosaccharides by Interference Phenomena in NMR Relaxation

Leila Ghalebani

Stockholm University

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Doctorial Dissertation 2008

Department of Physical, Inorganic and Structural Chemistry Stockholm University

ISBN 978-91-7155-714-8

Printed in Sweden by US-AB, Stockholm 2008

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Probing Dynamics of Oligosaccharides by Interference Phenomena in NMR Relaxation

Akademisk avhandling som för avläggande av filosofie doktorsexamen vid Stockholms Universitet offentligen försvaras I Magnélisalen, KÖL, Frescati,

fredagen den 12 september 2008, kl. 10.00 av

Leila Ghalebani

Avhandlingen för Fysikalisk Kemi Stockholm 2008 Stockholms Universitet ISBN 978-91-7155-714-8

Abstract

Oligosaccharides (carbohydrates) are a large class of biological molecules that are important as energy sources in the human body and have enormously varied biological functions. It is generally believed that biological activities of carbohydrates are related to their internal dynamics. The dynamic properties of some oligosaccharides in solution are studied in this thesis, by NMR relaxation. We have employed relaxation interference effects to investigate the conformational dynamics within oligosaccharides (intramolecular dynamics) and paramagnetic relaxation enhancement (PRE) as an experimental tool to study intermolecular dynamics. Most of the thesis concerns the dynamics of the methylene group in the two possibly mobile parts of the oligosaccharide: in the exocyclic hydroxymethyl moiety and in the glycosidic linkage position. To perform conformational dynamic studies, the more traditional auto-relaxation parameters are combined with the relaxation interference terms or the cross-correlated relaxation rates (CCRRs). Some experimental schemes based on the initial-rate technique were developed for measuring CCRRs. The techniques are useful for labelled sugars as well as naturally abundant ones. Furthermore, various dynamical models ranging from the Lipari–Szabo approach to several more informative and complicated models such as the two-site jump model, restricted internal rotation and slowly relaxing local structure (SRLS), have been employed to interpret our experimental data. We have combined and compared different models; we have also developed a novel approach to existing models, by scaling dipolar coupling constants (DCC), to extract the dynamic behaviour and structural properties of the system. We found that the auto- and cross-correlated relaxation data analyses yield a consistent picture of the dynamics in all cases. Additionally, our investigations show that CCRRs are practically important for verifica- tion of certain dynamical and structural information that is difficult to be determined by other means.

Moreover, the anisotropy of the carbon-13 chemical shielding tensor in the methylene group has been estimated, using the interference between dipole-dipole and chemical shift anisotropy.

This thesis also discusses using the PRE to investigate sugar dynamics relative to a paramagnetic MRI contrast agent in solution, which might be important in medicine. We have studied the intramolecular dynamics of the trisaccharide raffinose in the presence of a gadolinium complex. We also investigated the effect of translational diffusion instead of rotational diffusion, which is normally more important in NMR. The paramagnetically enhanced spin–lattice relaxation rates of aqueous protons over a wide range of magnetic fields and of carbon-13 and protons of the sugar at high fields have been measured. The nuclear magnetic relaxation dispersion of water protons and the PREs of proton and carbon in the sugar are interpreted in terms of the model recently developed in our laboratory, allowing both outer- and inner- sphere PREs for water protons, but allowing only the outer sphere PRE for nuclei in the sugar. We found that the relative diffusion has a stronger effect on the PRE than the electron spin relaxation.

Keywords: NMR, sugars, oligosaccharides, carbon-13 relaxation, cross-correlated relaxation, mo- lecular dynamics, internal motion, random jumps, PRE, paramagnetic relaxation

Department of Physical, Inorganic and Structural Chemistry

Stockholm University, 10691 Stockholm

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To my family Mansour & Romina

And in memory of my parents

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List of papers

The papers that this thesis is based on are the following:

I. Internal dynamics of hydroxymethyl rotation from CH2 cross- correlation in methyl-β-glucopyranoside

Katalin Köver, Gyula Batta, Jozef Kowalewski, Leila Ghalebani and Danuta Kruk Journal of Magnetic Resonance 2004; 167: 273-281 II. Cross-correlated and conventional dipolar carbon-13 relaxation in

methylene groups in small, symmetric molecules.

Leila Ghalebani, Piotr Bernatowicz, Sahar Nikkhou Aski and Jozef Kowalewski Concepts in Magnetic Resonance, Part A: Bridging Edu- cation and Research 2007; 30A (2), 100-115.

III. NMR relaxation interference effects and internal dynamics in γ- cyclodextrin

Leila Ghalebani, Dmytro Kotsyubynskyy, and Jozef Kowalewski Journal of Magnetic Resonance, in press

IV. NMR relaxation study of the motional dynamics of methylene groups in oligosaccharides by applying Slowly Relaxing Local Structure model

Leila Ghalebani, Dmytro Kotsyubynskyy, Jozef Kowalewski, Ulrika olsson and Göran widmalm, in manuscript

V. Nuclear spin relaxation study of aqueous raffinose solution in the presence of a gadolinium contrast agent

Leila Ghalebani, Danuta Kruk and Jozef Kowalewski Magnetic Resonance in Chemistry 2005; 43: 235-239

Papers I, II and V are reprinted by kind permission of the publishers.

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Introduction

The most abundant families of biomolecules in nature are carbohydrates, or saccharides (taken from the Greek word σάκχαρον for sugar), in addition to three other classes including proteins, lipids and nucleic acids. Carbohy- drates are the ultimate source of most of our food, clothing and even shelter.

They are also a major part of cellular structure and surfaces in animals and plants^{1, 2}. They have numerous functions in living systems; among others they are important in the immune system^3-5, cellular recognition^{6, 7}, regulation of protein function⁶, fertilization, blood clotting and develop- ment⁸. Oligosaccharides are composed of limited numbers of monosaccharide units, joined by glycosidic linkages. Characterization of the structures and dynamics of this large group of biomolecules is one of the most exiting research areas of chemistry. The determination of dynamic properties of carbohydrates plays a central role in the study of their biological activity by providing information on the type of activity of the molecule, and the way it changes upon binding with other molecules. It seems that the precise description of carbohydrate structure and dynamics at the molecular level is very essential to understanding their biological functions and their chemical behaviour in reactions⁷.

Nuclear magnetic resonance (NMR) spectroscopy is one of the most powerful and informative tools to investigate carbohydrate chemistry.

Nowadays one- and two-dimensional NMR techniques provide unique information sources for small molecules, like anomeric configuration of monosaccharides, and for large molecules, like the sequences of monosaccharide residues in oligo- and polysaccharides. Despite much recent progress in the field of protein dynamics^9-12, determination of internal dynamics of carbohydrates remains an arduous task, partly due to difficulties in isotope labelling of carbohydrates and partly because they have much more complex dynamics and are more difficult to study than proteins and nucleic acids.

However, because of technical progress in high-field magnets and develop- ment of more informative theoretical models, the quantitative understanding of dynamics of oligosaccharides has been improved during recent decades^13-

19. In NMR, the dynamics of carbohydrates can be probed by measurement of “nuclear spin relaxation” parameters. NMR relaxation is an extremely valuable investigative tool used extensively in both dynamic and structure determination at the molecular level. Relaxation parameters are useful to monitor the local and overall dynamics of sugars. Especially ¹³C relaxation experiments that can probe motion at several carbon sites of a molecule simultaneously have proved to be of importance.

In principle, the NMR relaxation parameters are sensitive to a very wide range of motional time scales, covering from sub-picoseconds to seconds^20-23 and several relaxation techniques with appropriate time scale are necessary

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Introduction

for the analysis of the various behaviours of carbohydrates in solution. The dynamic processes occur on different time scales, relative to molecular tumbling, including the internal motions faster and those that are comparable or even slower than the molecular tumbling. There are dynamic processes over a spectrum of time scales in carbohydrates, ranging from very rapid internal motion of some picoseconds and fast rotational diffusion of molecules and some internal motions of nanosecond time scales to a possible very slow conformational exchange that occurs in the micro- or even milli- second range. The latter dynamic, i.e. the possible slow processes, may be investigated by a T2 dispersion method that falls outside of this thesis. The very rapid dynamics has mainly been studied through “conventional”

relaxation measurements (spin–lattice relaxation time, spin–spin relaxation time and cross-relaxation or nuclear Overhauser enhancement, NOE) of various nuclei. To study the somewhat slower internal dynamics (still faster than overall tumbling), taking advantage of relaxation interference seems to be essential. Measurements of the conventional relaxation parameters are associated with studying build-up or decay of single-spin order, and have been customary since the introduction of NMR. The experimental develop- ments of studying multi-spin order, however, remained a less explored area of NMR relaxation until the recent few years.

When we speak about the build-up or decay of multi-spin orders, we deal with cross-correlated relaxation or interference effects. In terms of different mechanisms that cause nuclear spin relaxation, one deals here with interference between mechanisms. Besides the more traditional relaxation due to single mechanisms (e.g. dipole–dipole interactions), cross-correlated relaxation rates (CCRRs) that come from the interference between two mechanisms, such as dipole–dipole/dipole–dipole (DD/DD) or chemical shift anisotropy/dipole–dipole (CSA/DD) interactions, might give a deeper insight into the internal motions. The advantage of measuring cross-correlation or interference between two mechanisms is obtaining a clearer picture of the complicated dynamics and structures of molecules. Because of the small CSA value of aliphatic carbon, the contribution of the CSA mechanism can safely be neglected in conventional relaxation studies, while the interference terms are rather informative and one can determine the ¹³C anisotropy of chemical shift from (DD/CSA) interactions¹⁴. From the experimental point of view, the cross-correlations lead to differential line broadening, DLB, and also to different relaxation rates of multiplet lines. These differences can be measured by different techniques and used as a new tool for dynamical investigations.^{24, 25}

All NMR relaxation observables depend on dynamic properties of the spin system. Appropriate theoretical models of molecular dynamics are required to interpret experimental relaxation data in terms of dynamical properties of the system under investigation. Since most chemical reactions in biological and chemical systems proceed in solution, it has always been

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Introduction

interesting to develop models for describing the molecular structure and dynamics in liquids. These models help us to understand the relative position of the molecules and their motion and thus their reactivity in the solution.

The overall tumbling of sugars has often been considered isotropic, and the internal motion has typically been studied by Lipari–Szabo model free approach^26-28or some others. The usual theoretical models which are used for the study of the carbohydrate dynamics are based on the separation of the global tumbling from internal motion. Since this separability has been uncer- tain in the case of carbohydrates, for the first time we suggested employing the Slowly Relaxing Local Structure (SRLS)^29-32 model in this study to cover the internal motion coupled with overall tumbling.

In this thesis, I have proved experimentally that the study of relaxation assisted magnetization and coherence transfer in multi-spin systems expands the informational content of conventional nuclear spin relaxation studies. We developed an initial-rate experimental method for naturally abundant carbohydrates. In addition, we used recent methodological advances and modification of existing models, in the interpretation of relaxation data, which led to better understanding of the global dynamics, the separability of internal and overall motions and the presence of correlated dynamics as well as the relationship between reorientational dynamics and internal motion in different sugars.

Very rapid dynamics of oligosaccharides have widely been investigated by multiple fields and multiple temperatures carbon-13 “conventional”

relaxation data in our laboratory and others13, 16-19, 33, 33-39. The interest in this thesis is to extend our experimental and analyzing methods to study somewhat slower internal dynamics. It is shown that a combination of CCRRs and the “conventional” relaxation parameters provides a valuable possibility of obtaining a clear picture of the more complicated structural and dynamical properties of oligosaccharides, especially conformational jumps. In papers I-IV we deal with using the conventional relaxation parameters as well as interference terms to investigate the three-spin systems. We concentrate on the dynamics of a methylene group of a monosaccharide in paper I, and the structural information on several molecules in paper II. In this latter article we developed a simple experimental method and used some well known systems as models for testing it. A simple proton-coupled ¹³C inversion–

recovery experiment and a coupled spin-lock experiment have, respectively, been used for longitudinal and transverse CCRR measurements in the sys- tem. The molecules under investigation in paper II are small, rigid systems with a well-known structure: a spherical top system, hexamethylenetetra- mine (HMTA), a symmetric top system, quinuclidine and an approximately symmetric top system, 1-adamantanecarboxylic acid (AdCA). The conventional relaxation parameters as well as CCRRs data are interpreted in terms of spectral densities obtained for a rigid body diffusing in small steps.

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Introduction

Analyzing the results is promising with regard to the interpretation of auto- and cross-correlated relaxation parameters using these models.

In papers III and IV, we employed the experimental method developed in paper II to measure DD/DD and CSA/DD CCRRs in two oligosaccha- rides. We investigated the dynamics of methylene groups in γ-cyclodextrin and a disaccharide β-D-Glucopyranoside-(1→6)-α-D-Man-OMe. The interpretation of experimental data in terms of models differs in the two molecules. In γ-cyclodextrin, which is a conic oligosaccharide involving eight glucose residues, with the methylene groups located on top of the symmetric system, several models fit the experimental data well. In contrast, the methylene group in a disaccharide is located in a glycosidic linkage position (in the backbone) and experiences more complicated motion, so the relaxation data do not fit the tested models for a carbohydrate. We tried different models, and from the result of fitting we concluded that the internal motions are as slow as the overall motion, so that the two motions are not statistically independent. Therefore we suggested applying the SRLS model for interpreting this coupled motion in the disaccharide. We also used γ- cyclodextrin as a test molecule: SRLS fully fits the γ-cyclodextrin experi- mental data. For the disaccharide, the fitting is not consistent when we in- clude transverse CCRRs in addition to four other types of experimental data.

One possible reason for this problem may reside in the potential we assumed, the potential of the methylene group has been assumed axially symmetric in both oligosaccharides. We expect more consistent fitting for the disaccharide by applying a rhombic potential. This is the future plan in paper IV, which is included in this thesis as manuscript.

The final article deals with sugar dynamics in the presence of a paramag- netic contrast agent. The contrast agents are used to provide improved images in magnetic resonance imaging, MRI. They are injected before or during MRI to help diagnose problems or diseases of the brain or spine. One type of chemical contrast agent is a complex of a paramagnetic metal ion such as gadolinium (Gd). The Paramagnetic Relaxation Enhancement (PRE) effect is used to investigate the behaviour of the trisaccharide raffinose in the presence of a Gd complex in paper V. The water proton Nuclear Magnetic Relaxation Dispersion (NMRD), together with PRE measurements for the carbon-13 nuclei and protons of raffinose are used. We proved that the full set of PRE data can be interpreted consistently, using the theoretical tools developed by Kruk and Kowalewski⁴⁰.

The thesis outline is as follows: in the first chapter, the general nuclear spin relaxation theory is presented. In chapters 2 and 3, nuclear spin relaxation in coupled-spin and paramagnetic systems are outlined, respectively.

The methods used for measuring relaxation parameters are explained in chapter 4 and, after discussion of the papers on which this thesis is based in chapter 5, the conclusions is drawn in chapter 6. Finally, acknowledgments are made and the related articles are enclosed.

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1.1 Theory, general aspects

1 Nuclear Spin Relaxation

1.1 Theory, general aspects

The process of returning a perturbed system to its equilibrium is known as

“relaxation”. In equilibrium, the populations of the eigenstates have been rebuilt to the Boltzmann distribution, and all coherences have vanished.

Relaxation in liquid state NMR originates from random processes that influence interactions in the spin system. These are typically caused by molecular reorientation, internal motions or chemical exchange involving the nuclei of interest. The anisotropic spin interactions are averaged out to the first order as an effect of the motion, and therefore they do not appear in the fine structure of the spectrum, while the time dependence of anisotropic interactions contributes to relaxation. The fluctuations of local fields are sensitive to internal motions, and therefore we are able to extract dynamic information. From a macroscopic point of view, the return to equilibrium can be characterized by relaxation rates, which explain the build-up (recovery of z-magnetization to its thermal equilibrium in the case of longitudinal relaxa- tion) or decay (decay of magnetization perpendicular to the z axis to zero in the case of transverse relaxation) of the detected signal as a function of time.

Interpretation of NMR relaxation parameters, such as spin–lattice and spin–spin relaxation rates, cross-relaxation or NOEs and cross-correlated relaxation rates etc in terms of spectral densities of these motions, J (ω), yields dynamic and structural information about the molecules. Basically, the spectral density of molecular motion can be defined in terms of correlation times, and any NMR relaxation parameter can be interpreted as a function (usually as a linear combination) of spectral densities at different frequencies, and depend on the various contributing relaxation mechanisms.

Relaxation theory is well described in several books^{23, 41-43} and review articles^44-47. In the theory of nuclear spin relaxation, the purpose is to derive equations that correctly describe the experimental results. To do this, one should suggest a Hamiltonian for the system under study, solve the Liouville–von Neumann equation for the time evolution of the density operator, compute the expected time evolution of the experimental parameters and compare (least-squares fitting) the result with measured values of these observables. Then, one will be able to extract the dynamic and structural properties of the system of interest from these relaxation parameters.

In 1946, Bloch for the first time presented the theory of nuclear spin relaxation⁴⁸. Bloch’s equations of motion are suitable for predicting the behaviour of an ensemble of isolated spins interacting weakly with the lattice.

Briefly, the phenomenological Bloch equations give the time constants for the evolution of the longitudinal and transverse spin magnetization to their

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1 Nuclear Spin Relaxation

equilibrium: the spin-lattice relaxation time (T1) and the spin-spin relaxation time (T2)^48-50, respectively. T1-1 is the rate at which the average z- magnetization of non-interacting spins returns to equilibrium, and T2-1 is the rate of xy-magnetization (coherences) decay to equilibrium. Later on, Solo- mon⁵¹ suggested his equations for relaxation of a two-spin system perturbed by dipole–dipole interaction. Under conditions in which groups of nuclear spins couple with each other, a more complete treatment is required. In the mid-nineteen-fifties the fundamental theory of nuclear spin relaxation in scalar coupled spin system was formulated in the form of the Bloch–

Wangsness–Redfield master equation^{52, 53}, which is called perturbation, Bloch–Wangsness–Redfield or Redfield theory^{44, 54}.A full theoretical description of the spin relaxation using Redfield relaxation theory can be found in the book by Abragam (the operator representation)⁴¹ and the book by Slichter (the matrix representation)⁴². The Redfield relaxation theory is applicable to any type of non-equilibrium system and any relaxation mechanism. It can also be expressed in the Liouville superoperator formalism. Using this formalism is necessary in the case of slow-motion conditions in paramagnetic relaxation theory²³. I shall try to outline the usage of Red- field or perturbation theory in the explanation of relaxation of a three-spin system (methylene groups).

1.2 The Master Equation of Relaxation

In the semi-classical theory of spin relaxation, where the spins are treated quantum mechanically and the lattice is treated classically, one considers a spin system with a Hamiltonian consisting of a time-independent part, Hˆ₀or the main Hamiltonian, that acts only on the spin system, and a stochastic time-dependent part (the perturbation Hamiltonian), H ′ˆ t(), that couples the spin system to the lattice:

( ) 0 ( )

H t =H +H t′ (1.1) The perturbation Hamiltonian represents the interaction between spin states and variation of the interactions by random motions. The state of the spin system is described by the density matrix, σ (t). The time evolution of the density matrix is described by the Liouville–von Neumann equation, which can be written as (the notations are based on Kumar et al.⁵⁵):

ˆ0 ˆ

( ) [ ( ), ( )]

d t i H H t t

dtσ = − + ′ σ (1.2)

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1.2 The Master Equation of Relaxation

This equation (without going into details) can be written in matrix form by means of the widely used Redfield equations as:

( ) ( ) ( ( ) 0)

d t i t R t

dt ^αα ^αα ^αα ββ ^{αα ββ} ^ββ

σ ′ ω σ′ ′ ′ ′σ σ ′

′

= − +

∑

− (1.3)

The density matrix is expressed in the basis of eigenvectors of the unperturbed Hamiltonian,Hˆ₀, so that _ω_α_α_′=_ω_α−_ω_α_′ is the frequency of the transitionα→α′and R_α_α_′_β_β_′ is the real part of the relaxation matrix. The elements R_α_α_′_β_β_′ are expressed in the form:

( ) ( ) ( ) ( )

R_{αα ββ} J_{αβα β} _αβ J_{αβα β} _{α β} _{α β} J_γβγα _γβ _αβ J_{γα γβ} _γβ

γ γ

ω ω δ ω δ ω

′ ′= ′ ′ + ′ ′ ′ ′ − ′ ′

∑

−

∑

′ ′ ′ (1.4)

where the spectral densities are the Fourier transforms of the correlation function and are defined as:

( ) Re 0 ( ) exp( )

J_{αβα β}′ ′ω =

∫

^∞G_{αβα β}′ ′τ −iωτ τd (1.5) The random processes are described by the time correlation function,

) (τ

β α αβ ′′

G , which contains information about the molecular motion of the system and is given by:

( ) ( ) ( )

G_{αβα β}_{′ ′}τ =α H t′ β α′H t′ −τ β′ (1.6) Here, the bar represents an ensemble average. The perturbation Hamiltonian is composed of a number of distinct relaxation mechanisms, (r), for instance dipolar interactions between pairs of spins or an anisotropic chemical shift of the spin of interest:

( ) r( )

r

H t′ =

∑

H t′ (1.7) The correlation function will then contain several auto- and cross-correlation terms given by:

( ) _r( ) _r( ) _r( ) _r( )

r r r

G_{αβα β}′ ′τ α H t β α H t τ β α H t β α H′t τ β

≠′

′ ′ ′ ′ ′ ′ ′ ′

=

∑

− +

∑

− ^(1.8)

The first term on the right hand side is the auto-correlation term, and the second term is the cross-correlation term.

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1.3 Relaxation Hamiltonian

For various relaxation mechanisms the perturbation Hamiltonian can be expanded into a sum of products of time-invariant spin operators and time- dependent spatial operators. The Hamiltonian can be expressed in a compact form, in terms of irreducible tensor operators (appendix A):

2

( ) ( )

2

( ) ^r ( )^qˆ_r^q _r ^q( )

r q

H t ξ A F⁻ t

=−

′ =

∑ ∑

− (1.9) Where F_r⁽⁻^q⁾(t) is a random function of spatial variables (or the classical lattice function), generally depending on molecular motions, A_r^(q⁾ is a tensor spin operator, and ξ^r is the relevant interaction strength constant, which will be defined later. In order to describe the rotational behaviour of molecules, which makes the relevant interaction time-dependent, one needs to carry out a series of transformations from the molecular frame to the lab frame.

Irreducible tensors transform under rotation in the same way as spherical harmonics, and therefore it is more convenient to express the Hamiltonian in terms of irreducible tensors.

The intramolecular direct dipole–dipole interaction is the dominant and by far the most informative relaxation mechanism for a proton-bearing ¹³C.

TheF⁽⁻^q⁾( )t as well as the A^{( )}^q are rank-two tensors; and the spin operators of the dipolar interaction between two spins A and Xare given by⁵⁶:

2

1

0

( )

[4 ( )] / 6

A X

z z

A X A X

z z

A X A X A X

A I I

A I I I I

A I I I I I I

±

± ± ±

+ − − +

=

= ± +

= − +

∓ ∓

(1.10)

While the space part, F⁽⁻^q⁾( )t , is proportional to the inverse of the third power of the distance between two spins; it is simply the spherical harmonic

2^q( , )

Y θ ϕ multiplied by some constants.

12 3

2

6 ( , )

5

q q

A X AX

F π r Y

γ γ ⁻ θ ϕ

 

= −  

  (1.11)

Here, γ_A, γ_X are the gyromagnetic ratios of the involved nuclei, r_AX is the distance between two interacting spins, A and X, and Y₂^q(θ,ϕ) are spherical harmonics of the second rank. θ and ϕ are the polar angles that define the orientation of the relevant relaxation vector with respect to a laboratory frame, defined so that the z axis coincides with the direction of the static magnetic field B₀. The spectral density is derived from the correlation function ⁽⁻⁾ via Fourier transformation.

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1.3 Relaxation Hamiltonian

( )

( ) Re 0 ( ) ( ) exp( )

q r r q q

rr r r

J ′ω =ξ ξ^′

∫

^∞ F t F′ ⁻ t−τ −iωτ τd (1.12) The polar angles of spherical coordinates and the orientation of the CH bond with respect to the laboratory frame are illustrated in Figure 1.1.

Figure 1.1: the polar angles θ and ϕ of spherical coordinates that specify the orientation of the CH vector

The local magnetic fields experienced by every nucleus can be affected by chemical shielding (the electronic environments). Since the chemical shielding is anisotropic (orientation dependent), the components of the local fields in the laboratory frame change with reorientation of the molecule.

These fluctuations of magnetic fields are a source of relaxation that is called the chemical shift anisotropy (CSA) mechanism. The CSA may become an important relaxation mechanism at higher magnetic fields for ¹³C spin relaxation. Especially in coupled-spin systems, the cross-correlations between CSA and DD are rich sources of dynamical and structural information. The spin irreducible tensors for CSA interaction A_CSA^(q⁾ can be expressed as follows:

2

1

0

0 (1/ 2) (2 / 6)Z

A

A I

±

± ±

=

∓ (1.13)

The space irreducible tensors contain spherical harmonics multiplied by a factor involving constants (related to the units and normalization) and the quantity γ_AB₀∆σ. Usually the chemical shielding tensor can be assumed

C

H

ϕ θ

x' z'

y'

Molecule-fixed frame

B₀

x z

y

Laboratory coordinate system

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axially symmetric, so that σ_xx=σ_yy≠σ_zz and ∆ =σ σ−σ⊥=σzz−¹2

(

σxx+σyy

)

is the shielding anisotropy.

Since other relaxation mechanism can safely be neglected for our spin system, I bring to the end the Hamiltonian saga and in the next chapter I will interpret the application of the theory of nuclear spin relaxation according to the Redfield formalism for coupled-spin systems.

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1.3 Relaxation Hamiltonian

2 Nuclear spin relaxation in coupled-spin systems

The dynamics of multiple-spin systems are significantly more complicated than those of an isolated spin-1/2 system. In an isolated-spin system, one is speaking about a two-level system in which the transition can be described by a single time constant, whereas in a multiple-spin system, where there are more than two energy levels, one cannot use the simple explanation of Bloch equations, and the theory becomes more complicated. The relaxation of the system is complex and can be characterized by the relaxation matrix given in Eq. (1.3). One can obtain more information about molecular motion in the case where two or more spins are coupled. Modulation of the dipole–dipole interactions is produced by the correlated motion of internuclear vectors. We speak about auto-correlation dipolar relaxation when the motion of each pair of spins correlates with itself and about cross-correlation relaxation when it correlates with other motions in the system (interference between two internuclear vectors).

The observation of different relaxation behaviour of the central and outer lines of ¹³C multiplet spectra during the early 1970s was the beginning of using interference effects between two dipolar interactions^{57, 58}. At a later time, by using stronger magnetic fields, the differences in relaxation behaviour of outer lines in the multiplet were discovered^59-61. The dipolar cross-correlation gives rise to the difference in relaxation behaviour between outer and inner lines, while the differences in relaxation behaviour among outer lines in the multiplet comes from the dipolar-CSA cross-correlation^57,

58, 62-64.

This thesis concerns the dynamics of weakly coupled spin-1/2 systems, which are common systems in solution NMR. In a weakly coupled spin system it is assumed that all the coupling constants are much smaller than the differences in chemical shifts. The well known product operator formalism provides a convenient tool for description of experiments in a weakly coupled spin system. There are a total of 64 possible product operators for a three-spin system, which seems to be a rather complicated system to deal with⁶⁵. However, in most cases only a small subset of operators is important.

A transformation to the basis of “modes”, many of which can be directly related to observables, seems to be useful. For longitudinal relaxation, these modes are known as “magnetization modes”. Magnetization modes have been applied with great success to explain longitudinal interference terms in weakly coupled systems^{56, 66}.

I will not explain the details of the mathematical basis for the equations of motion used in this thesis; the reviews by Kumar and co-workers⁵⁵ Werbelow and Grant⁶⁷, Vold and Vold⁶³, Canet⁶⁴ and Fisher⁶⁸ provide a comprehensive overview of the theoretical developments in coupled-spin relaxation and also a valuable compendium of equations for a variety of

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2 Nuclear spin relaxation in coupled-spin systems

coupled-spin cases. I provide a brief theoretical overview of the application of Redfield theory and using magnetization modes as basis set for three-spin systems, in laboratory and rotating frames, in the next two sections.

2.1 Longitudinal cross-correlated relaxation

The longitudinal relaxation refers to the recovery of observables to their equilibrium by the rate determined by diagonal elements of the density matrix values. According to the secular approximation, and in the absence of radiofrequency field, r.f., irradiation, it is possible to consider the time evolution of the longitudinal and transverse magnetization independently.

The time evolution of all the diagonal elements is in general coupled and, according to Eq. (1.3) is given by^{23, 55, 69}:

( ( ) 0)

d R t

dt

αα

ααββ ββ

β

σ =

∑

σ −σ (2.1)

where

2 ( ) 2 ( )

R_ααββ J_αβαβ _αβ _αβ J_γαγα _γα

γ

ω δ ω

= −

∑

(2.2) Eq. (2.1) is identical to the rate equation describing the recovery of the populations of various energy levels (Pα=σαα) to their equilibrium values

( )

Pα⁰

through the transition probability approach, written as^{41, 42, 55}: ( 0)

dP W P P

dt

α

αβ β β

β

=

∑

− (2.3) where W_αβ =R_ααββ are the transition probabilities, andW_αα=−

∑

_β≠_αW_αβ.

As I mentioned in the previous section, in a weakly coupled spin system it seems to be easier and more informative to use a certain basis set of observables, the so-called “magnetization modes”, instead of the eigenstates of the Zeeman Hamiltonian for describing the relaxation matrix^{64, 67}. Magnetization modes are essentially various linear combinations of the populations of different energy levels, which can be easily associated with observable quantities and which lead to simplifications of the kinetic equation describing the evolution of the system under longitudinal relaxation.

One defines single-spin magnetization modes, such asA_z, M_z, X_z, …, two- spin magnetization modes, 2A Mz z, 2A_zX_z,… and multi-spin modes up to N spins. Each mode represents the expectation value of the products of the

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2.1 Longitudinal cross-correlated relaxation

corresponding spin operators, for example A tz^{( )}= Iz^A ^{( )}t =Tr

{

σ^{( )}t Iz^A

}

and²^{A X t}^z ^z^{( )}⁼ ²^{I I}^z^A ^z^X ^{( )}^t ⁼^Tr

{

^σ^{( )2}^t ^{I I}^z^A ^z^X

}

^.

The dimensionality of the vector P(t) is in general 2^N, where N is the number of coupled nuclear spins, and there are as many magnetization modes (eight modes for a three-spin system i.e. as many as the number of eigenstates of _H^ˆ₀). The set of linear first-order coupled differential equations given by Eq. (2.3) may be further simplified using both the molecular symmetry of the coupled spins and the intrinsic spin inversion symmetry (chang- ing the signs of all the magnetic quantum numbers) that the Hamiltonian holds for all spin relaxation mechanisms. This symmetry feature yields sets of coupled differential equations with “symmetric” and “anti-symmetric”

dynamical variables. Thus for DD interaction, the maximum dimensionality of coupled differential equations for treating N spins can always be reduced to 2^N-1, and further reduction may be obtained if the molecular symmetry introduces degeneracies of the spin energy levels⁶⁷. Mathematically, these features of symmetry can be incorporated in an appropriate similarity transform as follows:

( )

,

( ) ( ) ^eq( )

i

ij j j

i j

d t dt

ν =

∑

Γ ν τ −ν τ (2.4)

where ν( )t =QP t( ) and Γ QWQ= ⁻¹. The matrix Q describes the transformation between the two basis sets. Selection of Q within the symmetric and anti-symmetric subsets is not arbitrary, but instead must specify those mini- mum combinations of level populations which completely span all of the independent observables of the experiment. This transformation has the advantage that all elements of ν have a real physical interpretation56, 64, 67. With this short introduction about magnetization modes, I turn to the relaxation matrix in the magnetization modes basis.

Starting with the simplest case, the single isolated-spin ensemble, the evolution of the longitudinal magnetization is described by means of the Bloch equation as ⁴⁸:

ˆ_z / 1( ˆ_z ˆ_eq)

d I dt= −R I −I (2.5) where R1 is the longitudinal relaxation rate and I^ˆ_eq is the equilibrium mag- netization. For an ensemble of two spin systems, let us say A and X, the relevant longitudinal relaxation processes are no longer monoexponential, and their evolution can be described according to the Solomon equation ⁵¹:

ˆ ˆ

A A A

z A AX z eq

X X X

AX X

z z eq

I I I

d

dt I I I

ρ σ

σ ρ

    − 

 = −  

   

    − 

   

(2.6)

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2 Nuclear spin relaxation in coupled-spin systems

whereρA and ρX are the longitudinal relaxations rates for spins A and X, and σAX is known as the cross-relaxation rate.

The three-spin system is sufficiently complex to illustrate most of the principles governing coupled multiple-spin relaxation^{58, 70-73}, and yet a sufficiently common spin system (all ¹³C-containing methylene groups) to be of considerable importance. The three spins in the group form an AX2 or AMX system depending on the chemical and/or magnetic equivalence of the two protons. The energy level diagram and a possible set of basis functions of a weakly-coupled three-spin system, AMX (A=¹³C, M, X=¹H), is depicted in Figure 2.1.

Figure 2.1: The energy-level diagram (based on Kumar et al.⁵⁵) and a basis set of weakly coupled three- spin systems AMX (A=¹³C, M, X=¹H). α and β correspond to Zeeman eigenstates; their product repre- sents the different eigenstates of the three-spin system, numbered from 1 to 8, and λ is the total spin inversion operator. The first spin state in wave-function k corresponds to the ¹³C spin state, and the remaining spin states are associated with the proton spins. The dashed lines represent the four single quantum transitions of the A (carbon-13) spin, the solid and dash-dot lines are for M and X spins (the two protons), respectively.

There are four transitions for spin A in the AX2 or AMX system; a triplet will be observed instead of quartet due the fact that the difference between the two scalar coupling (J-coupling) constants is small compared to the line width. Since protons are chemically or magnetically non-equivalent in most cases in this study, let me consider the basis set for the AMX system to describe the longitudinal and transverse relaxation (relaxation in the presence of an r.f. field) equations. The unitary transformation for moving from population to the magnetization modes in an AMX spin system is given by:

1 , 8 1

2 , 7 2

3 , 6 3

4 , 5 4

ααα βββ λ

ααβ ββα λ

αβα βαβ λ

βαα αββ λ

≡ ≡ =

Probing Dynamics of Oligosaccharides by Interference Phenomena in NMR Relaxation

Probing Dynamics of Oligosaccharides by Interference Phenomena in NMR Relaxation

Probing Dynamics of Oligosaccharides by Interference Phenomena in NMR Relaxation

Abstract

Contents

List of papers

Introduction

1 Nuclear Spin Relaxation

1.1 Theory, general aspects

1.2 The Master Equation of Relaxation

∑

∑

∑

∫

∑

∑

∑

1.3 Relaxation Hamiltonian

∑ ∑

∫

(

)

2 Nuclear spin relaxation in coupled-spin systems

2.1 Longitudinal cross-correlated relaxation

∑

∑

( )

∑

∑

{

}

{

}

( )

∑